The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.
Mots clés : spectral analysis, dimension reduction, periodic homogenization, Γ-convergence, asymptotic expansions
@article{COCV_2012__18_2_427_0,
author = {Ferreira, Rita and Mascarenhas, Lu{\'\i}sa M. and Piatnitski, Andrey},
title = {Spectral analysis in a thin domain with periodically oscillating characteristics},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {427--451},
publisher = {EDP-Sciences},
volume = {18},
number = {2},
year = {2012},
doi = {10.1051/cocv/2011100},
mrnumber = {2954633},
zbl = {1248.35135},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011100/}
}
TY - JOUR AU - Ferreira, Rita AU - Mascarenhas, Luísa M. AU - Piatnitski, Andrey TI - Spectral analysis in a thin domain with periodically oscillating characteristics JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 427 EP - 451 VL - 18 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011100/ DO - 10.1051/cocv/2011100 LA - en ID - COCV_2012__18_2_427_0 ER -
%0 Journal Article %A Ferreira, Rita %A Mascarenhas, Luísa M. %A Piatnitski, Andrey %T Spectral analysis in a thin domain with periodically oscillating characteristics %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 427-451 %V 18 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011100/ %R 10.1051/cocv/2011100 %G en %F COCV_2012__18_2_427_0
Ferreira, Rita; Mascarenhas, Luísa M.; Piatnitski, Andrey. Spectral analysis in a thin domain with periodically oscillating characteristics. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 427-451. doi : 10.1051/cocv/2011100. http://www.numdam.org/articles/10.1051/cocv/2011100/
[1] and , Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | MR | Zbl
[2] and , Analyse asymptotique spectrale d'un problème de diffusion neutronique. C. R. Acad. Sci. Paris, Ser. I 324 (1997) 939-944. | MR | Zbl
[3] , Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984). | MR | Zbl
[4] and , Homogenization of Processes in Periodic Media. Nauka, Moscow (1984). | MR | Zbl
[5] , and , Asymptotic analysis for periodic structures. North-Holland Publishing Co., Amsterdam (1978). | MR | Zbl
[6] , and , On the curvature and torsion effects in one dimensional waveguides. ESAIM : COCV 13 (2007) 793-808. | Numdam | MR | Zbl
[7] , An introduction to Γ-convergence, Progress in Nonlinear Differential Equations and their Applications 8. Birkhäuser Boston Inc., Boston (1993). | MR | Zbl
[8] and , Waves in a thin and periodically oscillating medium. C. R. Math. Acad. Sci. Paris, Ser. I 346 (2008) 579-584. | MR | Zbl
[9] and , Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, New York (1977). | MR | Zbl
[10] , and , Homogenization of differential operators and integral functionals. Springer-Verlag, Berlin (1994). | MR | Zbl
[11] , Homogenization of Elliptic Eigenvalue Problems : Part 1. Appl. Math. Optim. 5 (1979) 153-167. | MR | Zbl
[12] , Homogenization of Elliptic Eigenvalue Problems : Part 2. Appl. Math. Optim. 5 (1979) 197-216. | Zbl
[13] and , Effective diffusion for a parabolic operator with periodic potential. SIAM J. Appl. Math. 53 (1993) 401-418. | MR | Zbl
[14] and , Degeneration of effective diffusion in the presence of periodic potential. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 571-587. | Numdam | MR | Zbl
[15] and , H-Convergence, in Topics in the mathematical modelling of composite materials. Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston (1997). | MR | Zbl
[16] , and , Mathematical problems in elasticity and homogenization. Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam (1992). | MR | Zbl
[17] , Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math Sci. 90 (1981) 239-271. | MR | Zbl
[18] and , Regular degeneration and boundary layer for linear differential equations with small parameter. Amer. Math. Soc. Transl. (2) 20 (1962) 239-364 [English translation]. | MR | Zbl
Cité par Sources :
